Optical fiber and optical transmission system

ABSTRACT

An optical fiber includes a core, a first clad that is provided on an outer circumferential portion of the core and has a refractive index lower than that of the core, and a second clad that is provided on an outer circumferential portion of the first clad and has a refractive index lower than that of the first clad. In the optical fiber, a mode field diameter at a wavelength of 1.55 μm is equal to or greater than 11.5 μm, a cutoff wavelength is equal to or less than 1.53 μm, a bending loss at a bending radius of 30 mm and a wavelength of 1.625 μm is equal to or less than 2.0 dB/100 turns, and a delay time of transmission light per unit length at a wavelength of 1.55 μm is equal to or less than 4.876 μs/km.

TECHNICAL FIELD

The invention relates to an optical fiber and an optical transmissionsystem. Priority is claimed on Japanese Patent Application No.2017-130725, filed Jul. 3, 2017, the content of which is incorporatedherein by reference.

BACKGROUND ART

With recent diversified use of communication networks, there is a demandfor a reduction of transmission delay. For example, in communicationbetween computers which is frequently used for financial transactionswhich are carried out on an international scale, a reduction in atransmission delay of 1 ms has a great influence on the transmissionperformance of communication, financial transaction services, andcustomer profit or loss. In the future, the demand for further reductionof a transmission delay is expected to accelerate.

In long-distance communication networks such as a submarine opticalcable network crossing the Pacific Ocean, a construction length of acommunication line amounts to several thousands of kilometers. Forimprovement in transmission performance of a long-distance communicationnetwork, it is important to reduce a delay which occurs in atransmission line. In a submarine optical cable network, effort toreduce a delay in a communication line by optimizing a constructed routeof an intercontinental submarine optical cable network are made. By thiseffort, it has been reported that a transmission delay of along-distance communication network is reduced by about several ms.

A delay of a communication network includes a delay which is caused in adevice such as a transmission device and a delay which is caused in atransmission line.

In a long-distance communication network, a delay time which is causedin a transmission line occupies most of a delay time which occurs in thewhole network and becomes too long to ignore.

A delay time of an optical fiber per unit time in a transmission line ofa communication network is mainly determined according to a refractiveindex of a medium of the optical fiber. In order to reduce atransmission delay of a communication network, it is effective to use amedium with a low refractive index as a medium of an optical fiber. Acutoff shift fiber which is used for a submarine optical cable networkin the related art includes a core which is formed of silica glass withhigh purity. Accordingly, in a submarine optical cable network in therelated art, light can be transmitted with a delay time of about 4.876μs/km in a wavelength band including 1.55 μm. In Non-Patent Literature1, it is reported that a delay time of a photonic band gap fiber isreduced by about 3.448 μs/km. The photonic band gap fiber includes ahollow core in which a refractive index of a medium is lowered to theutmost limit.

CITATION LIST Non-Patent Literature

[Non-Patent Literature 1]

N. V. Wheeler et al., “Wide-bandwidth, low-loss, 19-cell hollow corephotonic band gap fiber and its potential for low latency datatransmission,” OFC/NFOEC Post deadline Papers PDP5A.2 2012.

SUMMARY OF INVENTION Technical Problem

As described above, in a long-distance communication network, a delaytime is improved by optimization of a constructed route. However,geometrical conditions or construction costs are limited in actuallyoptimizing a constructed route, and there is a problem in that areduction amount in delay time due to optimization of a constructedroute is decreased by this limitation.

In a photonic band gap fiber, a hollow core with a low refractive indexis formed. However, since a transmission loss of a photonic band gapfiber amounts to about several dB/km, there is a problem in that thephotonic band gap fiber is not suitable for a transmission line of along-distance communication network.

The invention is made in consideration of the above-mentioned problemsand an objective thereof is to provide an optical fiber that can beapplied to a long-distance communication network, has a mode fielddiameter (MFD) and a bending loss which are equivalent to an MFD and abending loss of a cutoff shifted fiber according to the related art, andhas a delay time which is less than a delay time of the cutoff shiftedfiber.

The invention provides an optical transmission system of which theoptical fiber has excellent characteristics.

Solution to Problem

The inventors newly found design conditions and structures of an opticalfiber having five features: (1) the optical fiber includes a core, afirst clad which is adjacent to an outer circumferential portion of thecore, and a second clad which is adjacent to an outer circumferentialportion of the first clad. (2) a radius of the core is equal to or lessthan 4 μm, (3) a relative refractive index difference of the first cladfrom the core is equal to or less than 0.0%, (4) a mode field diameter(MFD) at a wavelength of 1.55 μm is equal to or greater than 11.5 μm,and (5) a bending loss at a bending radius of 30 mm and a wavelength of1.625 μm is equal to or less than 2.0 dB/100 turns as an optical fiberfor solving the above-mentioned problems.

An optical fiber according to the invention includes a core, a firstclad that is provided on an outer circumferential portion of the coreand has a refractive index lower than that of the core, and a secondclad that is provided on an outer circumferential portion of the firstclad and has a refractive index lower than that of the first clad. Amode field diameter of the optical fiber according to the invention at awavelength of 1.55 μm is equal to or greater than 11.5 μm. A cutoffwavelength of the optical fiber according to the invention is equal toor less than 1.53 μm. A bending loss of the optical fiber according tothe invention at a bending radius of 30 mm and a wavelength of 1.625 μmis equal to or less than 2.0 dB/100 turns. A delay time of transmissionlight of the optical fiber according to the invention per unit length ata wavelength of 1.55 μm is equal to or less than 4.876 μs/km.

In the optical fiber according to the invention, a radius of the coremay be equal to or less than 1.0 μm and equal to or greater than 4.3 μm,and a radius of the first clad may satisfy Equations (1) and (2).

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack} & \; \\{\mspace{79mu} {a_{2} \geq {2\left\{ {\left( {1.43a_{1}^{- 1.45}} \right)^{2} - \left( {\Delta_{1} + {1.43a_{1}^{- 1.45}}} \right)^{2}} \right\}^{\text{?}}}}\;} & (1) \\{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack} & \; \\{{5.56 - {3.94\; {\log \left( {1 + \frac{\Delta_{1}}{0.19 + {0.69a_{1}^{- 2.00}}}} \right)}}} \leq a_{2} \leq {7.68 + {\left( {1.14 - {2.51\; a_{1}}} \right){\log \left( {1 + \frac{\Delta_{1}}{0.81a_{1}^{- 0.77}}} \right)}}}} & (2) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

In Equations (1) and (2), a₁ represents the radius [μm] of the core. a₂represents the radius [μm] of the first clad. At represents a relativerefractive index difference [%] of the first clad from the core.

In the optical fiber according to the invention, the relative refractiveindex difference of the second clad from the first clad may satisfyEquation (3).

$\begin{matrix}{{\Delta_{2} \geq {{- 0.033} + {\left( {{- 7.490} - {0.187a_{1}^{3.407}\Delta_{1}} - {0.044a_{1}^{4.324}\Delta_{1}^{2}}} \right)a_{2}^{{- 1.876} - {0.014\text{?}}}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (3)\end{matrix}$

In the optical fiber according to the invention, in a sectional view, aplurality of low-delay cores including the core as a first core and thefirst clad provided on an outer circumferential portion of the firstcore may be disposed on concentric circles at the center of the secondclad.

In the optical fiber according to the invention, the core may bedisposed as a second core at the center of the second clad.

In the optical fiber according to the invention, in a sectional view, alow-delay core including the core as a first core and the first cladprovided on an outer circumferential portion of the first core may bedisposed at the center of the second clad, and the core may be disposedas a third core on a concentric circle at the center of the low-delaycore.

In the optical fiber according to the invention, in a sectional view, alow-delay core including the core as a first core and the first cladprovided on an outer circumferential portion of the first core may bedisposed at the center of the second clad, and the cores may be packedmost closely as fourth cores around the low-delay core.

An optical transmission system according to the invention includes theabove-mentioned optical fiber, a transmitter that is connected to oneend of the optical fiber and a receiver that is connected to the otherend of the optical fiber.

Advantageous Effects of Invention

According to the invention, it is possible to provide an optical fiberthat has an MFD and a bending loss which are equivalent to an MFD and abending loss of a cutoff shifted fiber according to the related art andhas a delay time which is less than a delay time of the cutoff shiftedfiber. According to the invention, since compatibility between theoptical fiber and an existing submarine optical cable network isacquired, the delay time of the optical fiber decreases. According tothe invention, since a transmission line is constructed with the opticalfiber according to the invention, a delay time in the transmission lineof the optical transmission system decreases and a delay time in thewhole optical transmission system decreases.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a graph showing a relationship between optical characteristicsand a delay time in a silica-core fiber according to the related art.

FIG. 2 is a diagram showing a refractive index distribution of asingle-mode optical fiber (SMF) according to the invention.

FIG. 3 is a graph showing conditions of a radius a₂ and a relativerefractive index difference Δ₁ of an optical fiber according to theinvention in which all of a plurality of conditions in which a cutoffwavelength is 1.53 μm, an MFD is equal to or greater than 11.5 μm, abending loss is equal to or less than 2 dB/100 turns, and a delay timeis equal to or less than a delay time of a cutoff shift fiber aresatisfied when a radius of a core is 1.0 μm.

FIG. 4 is a graph showing conditions of the radius a₂ and the relativerefractive index difference Δ₁ of the optical fiber according to theinvention in which all of the plurality of conditions in which thecutoff wavelength is 1.53 μm, the MFD is equal to or greater than 11.5μm, the bending loss is equal to or less than 2 dB/100 turns, and thedelay time is equal to or less than the delay time of the cutoff shiftedfiber are satisfied when the radius of the core is 1.5 μm.

FIG. 5 is a graph showing conditions of the radius a₂ and the relativerefractive index difference Δ₁ of the optical fiber according to theinvention in which all of the plurality of conditions in which thecutoff wavelength is 1.53 μm, an MFD is equal to or greater than 11.5μm, the bending loss is equal to or less than 2 dB/100 turns, and thedelay time is equal to or less than the delay time of the cutoff shiftfiber are satisfied when the radius of the core is 2.0 μm.

FIG. 6 is a graph showing a relationship between a fitting coefficientof the delay time and a radius of a core of an SMF according to theinvention in which predetermined required conditions are satisfied.

FIG. 7 is a graph showing a relationship between a fitting coefficientof a MFD and the radius of the core of the SMF according to theinvention in which predetermined required conditions are satisfied.

FIG. 8 is a graph showing a relationship between a fitting coefficientof a bending loss and the radius of the core of the SMF in the inventionin which predetermined required conditions are satisfied.

FIG. 9 is a graph showing conditions of a radius a₂ and a relativerefractive index difference Δ₁ of an optical fiber according to theinvention in which all of a plurality of conditions in which a cutoffwavelength is 1.53 μm, an MFD is equal to or greater than 11.5 μm andequal to or less than 12.5 μm, a bending loss is equal to or less than 2dB/100 turns and a delay time is equal to or less than a delay time of acutoff shift fiber are satisfied when a radius of a core is 1.0 μm.

FIG. 10 is a graph showing conditions of the radius a₂ and the relativerefractive index difference Δ₁ of the optical fiber according to theinvention in which all of a plurality of conditions in which the cutoffwavelength is 1.53 μm, the MFD is equal to or greater than 11.5 μm andequal to or less than 12.5 μm, the bending loss is equal to or less than2 dB/100 turns and the delay time is equal to or less than the delaytime of the cutoff shift fiber are satisfied when the radius of the coreis 1.5 μm.

FIG. 11 is a graph showing conditions of the radius a₂ and the relativerefractive index difference Δ₁ of the optical fiber according to theinvention in which all of a plurality of conditions in which the cutoffwavelength is 1.53 μm, the MFD is equal to or greater than 11.5 μm andequal to or less than 12.5 μm, the bending loss is equal to or less than2 dB/100 turns and the delay time is equal to or less than the delaytime of the cutoff shift fiber are satisfied when the radius of the coreis 2.0 μm.

FIG. 12 is a graph showing a relationship between a fitting coefficientof a bending loss and a radius of a core of the SMF according to theinvention in which predetermined required conditions are satisfied.

FIG. 13 is a graph showing a relationship between the radius a₂ and therelative refractive index difference Δ₁ of the SMF according to theinvention in which the cutoff wavelength is equal to or less than 1.53μm.

FIG. 14 is a graph showing a relationship between fitting coefficientsκ₉ and κ₁₀ of the cutoff wavelength, a radius a₁ of a core and arelative refractive index difference Δ₁ of a first clad from a core areaof the SMF according to the invention.

FIG. 15 is a graph showing a relationship between fitting coefficientsκ₁₁ and κ₁₂ of the cutoff wavelength and the radius a₁ of the core ofthe SMF according to the invention.

FIG. 16 is a graph showing a relationship between a fitting coefficientκ₁₃ of the cutoff wavelength and the radius a₁ of the core of the SMFaccording to the invention.

FIG. 17 is a diagram showing a refractive index distribution of the SMFaccording to the invention having a low-index layer in a second clad.

FIG. 18 is a diagram showing an example of an optical transmissionsystem including the SMF according to the invention.

FIG. 19A is a sectional view of a first example of an optical fiberaccording to the invention including a plurality of cores.

FIG. 19B is a sectional view of a second example of an optical fiberaccording to the invention including a plurality of cores.

FIG. 19C is a sectional view of a third example of an optical fiberaccording to the invention including a single core.

FIG. 19D is a sectional view of a fourth example of an optical fiberaccording to the invention including a single core.

FIG. 20 is a diagram showing an example of an optical transmissionsystem using the optical fiber according to the invention.

FIG. 21 is a graph showing a relationship between a radius of a core anda Rayleigh scattering loss which is applied to an electric fielddistribution by the core and a first clad when a mode field diameter is11.5 μm in the optical fiber according to the invention.

FIG. 22 is a graph showing a relationship between a radius a₁ and aRayleigh scattering loss which is applied to an electric fielddistribution by a core and a first clad when a mode field diameter is15.0 μm in the optical fiber according to the invention.

FIG. 23 is a graph showing a relationship between a Rayleigh scatteringloss α_(R), an MFD, a group delay time of light which is transmitted ina core per unit length and the radius a₁ in the optical fiber accordingto the invention based on Recommendation G654.D.

FIG. 24 is a graph showing a relationship between a Rayleigh scatteringloss α_(R), an MFD, a group delay time of light which is transmitted ina core per unit length and the radius a₁ in the optical fiber accordingto the invention based on Recommendation G654.E.

FIG. 25 is a diagram showing a refractive index distribution of anoptical fiber according to the invention which has been manufactured bytrial.

FIG. 26 is a diagram showing results of measurement and evaluation ofoptical characteristics in the optical fiber according to the inventionwhich has been manufactured by trial, a conventional SMF and a CSF(various fibers).

FIG. 27 is a graph showing results of measurement of wavelengthdependency of an MFD in the various fibers.

FIG. 28 is a graph showing results of measurement of wavelengthdependency (a loss wavelength spectrum) of a transmission loss in thevarious fibers.

FIG. 29 is a graph showing plots with negative fourth power ofwavelengths in the loss wavelength spectrum in the various fibers andfitting lines with the plots.

FIG. 30 is a graph showing measurement results of incident light powerdependency of a phase shift amount based on a CW-SPM method in nonlinearcoefficient evaluation of the various fibers and fitting lines of themeasurement results.

FIG. 31 is a graph showing measurement results of the group delay timebased on an impulse response approach in the various fibers.

FIG. 32 is a graph showing measurement results of wavelength dependencyof the group delay time based on the impulse response approach in thevarious fibers.

DESCRIPTION OF EMBODIMENTS

Hereinafter, embodiments of the invention will be described withreference to the accompanying drawings. The embodiments described beloware examples of the invention, but the invention is not limited to theseembodiments. In the specification and the drawings, elements having thesame functions will be referred to by the same reference signs anddescription thereof will not be repeated.

A silica-core fiber according to the related art includes a core (whichmay be referred to as a core area) formed of silica glass (SiO₂) with ahigh purity of 99.8 wt % or more and has a conventional step index typerefractive index distribution. As is well known, in a step index typerefractive index distribution, a refractive index of a core thattransmits light and a refractive index of a clad (which may be referredto as a clad area) are uniform. A relationship between opticalcharacteristics (a radius a [μm] of a core formed of silica and arelative refractive index difference Δ [%] of a clad from the core) anda delay time in a silica-core fiber according to the related art isshown in FIG. 1. The refractive index of silica is 1.444377 at awavelength of 1.55 μm.

Optical characteristics of a cutoff shifted fiber which is used mainlyfor a submarine optical cable are prescribed in Recommendation G654.D ofInternational Telecommunication Union-Telecommunication StandardizationSector (ITU-T). In Recommendation G654.D, it is prescribed that a modefield diameter (MFD) of a cutoff shift fiber at a wavelength of 1.55 μmis equal to or greater than 11.5 μm and equal to or less than 15.0 μm.In Recommendation G654.D, it is prescribed that a bending loss of thecutoff shift fiber at a wavelength of 1.625 μm and a bending radius of30 mm is equal to or less than 2.0 dB/100 turns and that a cutoffwavelength of the cutoff shift fiber is equal to or less than 1.53 μm.

Optical characteristics of an optical fiber which is used for a landcore network of long-distance communication are prescribed inRecommendation G654.E of the ITU-T. In Recommendation G654.E, it isprescribed that the MFD of the optical fiber for the land core networkat a wavelength of 1.55 μm is equal to or greater than 11.5 μm and equalto or less than 12.5 μm. In Recommendation G654.E, it is prescribed thata bending loss of the optical fiber for the land core network at awavelength of 1.625 μm and a bending radius of 30 mm is equal to or lessthan 0.1 dB/100 turns and that the cutoff wavelength of the opticalfiber for the land core network is equal to or less than 1.53 μm.

A solid line (A) in FIG. 1 indicates a relationship between a radius a[μm] of a core and a relative refractive index difference Δ [%] of aclad from the core of a silica-core fiber of which an MFD is 11.5 μm asprescribed in Recommendation G654.D. A solid line (B) in FIG. 1indicates a relationship between a radius a and a relative refractiveindex difference Δ of a silica-core fiber of which a banding loss α_(b)is 2.0 dB/100 turns. A solid line (C) in FIG. 1 indicates a relationshipbetween a radius a of a core and a relative refractive index differenceΔ of a silica-core fiber of which a cutoff wavelength λ_(c) is 1.53 μm.

As indicated by an arrow in FIG. 1, an area in which conditionssatisfying the prescriptions of Recommendation G654.D, that is,conditions in which the MFD is equal to or greater than 11.5 μm, thebending loss α_(b) is equal to or less than 2.0 dB/100 turns, and thecutoff wavelength λ_(c) is equal to or greater than 1.53 μm, is an areain which a part in which the relative refractive index difference Δ islower than that of the solid line (A) (that is, a part in which therelative refractive index difference Δ is close to 0 and an upper partin the graph of FIG. 1), a part in which the radius a is greater thanthat of the solid line (B), and a part in which the radius a is lessthan that of the solid line (C) overlap. It can be seen that asilica-core fiber having a structure satisfying the hatched part in FIG.1 satisfies all the conditions required for an optical fiber for along-distance communication network and has a core with a radius of 4.4μm or more.

Dotted lines in FIG. 1 indicate relationships between the radius a andthe relative refractive index difference Δ in silica-core fibers inwhich a group delay time (GD) (which may be simply referred to as adelay time) is 4.861 μs/km, 4.87 μs/km, 4.873 μs/km, 4.876 μs/km and4.879 μs/km. In a silica-core fiber satisfying prescriptions ofRecommendation G654.E, a bending loss which is lower than that of asilica-core fiber satisfying the prescriptions of Recommendation G654.Dis required. An area satisfying the prescriptions of RecommendationG654.E changes toward a part in which the radius a is greater than thatof the hatched part in FIG. 1.

As can be seen from a position relative to the hatched part in FIG. 1, aminimum group delay time which can be achieved in a cutoff shifted fiberis 4.876 μs/km. It can also be seen that the group delay time which isachieved decreases as the radius a decreases. When the radius a is equalto or less than 4.4 μm, the bending loss increases and it is difficultto reduce the group delay time. A reason why it is difficult to reducethe group delay time is that the most electric field distribution oftransmission light is confined to the core and the speed of thetransmission light is dominantly determined according to a refractiveindex of silica-glass with high purity, which is a material of the core.In a silica-core fiber with a refractive index distribution other than astep index type which has been developed to enable simultaneousenlargement of the core and reduction in loss, an electric fielddistribution is confined well to the core and thus a delay time thereofis equal to that of a step index type silica-core fiber or less thanthat of the step index type silica-core fiber. An example of therefractive index distribution other than a step index type is a W-shapedrefractive index distribution.

A distribution of a refractive index n of a single-mode optical fiber(SMF) (an optical fiber) according to the invention is shown in FIG. 2.In the invention, an absolute value of a relative refractive indexdifference Δ is not used and the value is basically a negative value. Anoptical fiber according to the invention includes a core (r≤a₁), a firstclad (a₁<r≤a₂) and a second clad (r>a₂) in a direction overlapping theradius r from the center of a section crossing a longitudinal direction.That is, the first clad is provided on an outer circumferential portionof the core, and the second clad is provided on an outer circumferentialportion of the first clad. A refractive index n₁ of the core isequivalent to a refractive index (a refractive index n_(SrO2)=1.444377at a wavelength of 1.55 μm) of silica glass with high purity or equal toor less than the refractive index of silica glass with high purity. Arefractive index n₂ of the first clad is less than the refractive indexn₁. A refractive index n₃ of the second clad is less than the refractiveindex n₂.

In an SMF in which a radius at is small, a delay time is expected todecrease and confinement of transmission light to the core is achievedby the second clad. Particularly, confinement of light of a firsthigh-order mode having a characteristic in which an electric fielddistribution of light expands relatively has a small influence ontransmission of light in the core. Accordingly, by designing the radiusa₁ and a relative refractive index difference Δ₂ of the second clad fromthe first clad as will be described later, a cutoff wavelength of theSMF according to the invention is optimized. On the other hand, afundamental mode is affected by a refractive index distribution in thevicinity of the center of a section of an optical fiber. Inconsideration thereof, the MFD or the bending loss of the SMF accordingto the invention are appropriately set by adjusting the radii a₁ and a₂.

First Embodiment

A relationship between optical characteristics and a delay time in anSMF according to a first embodiment of the invention is shown in FIG. 3.A radius at of an SMF according to the first embodiment is set to 1.0μm. In consideration of a relative refractive index difference Δ₂ [%]and a radius a₂, a cutoff wavelength of the SMF according to the firstembodiment is set to be equal to or less than 1.53 μm. A solid line (D)in FIG. 3 represents a relationship between the radius a₂ and a relativerefractive index difference Δ₁ of the SMF according to the firstembodiment in which an MFD is equal to or greater than 11.5 μm. A solidline (E) in FIG. 3 represents a relationship between the radius a₂ andthe relative refractive index difference Δ₁ of the SMF according to thefirst embodiment in which a bending loss is equal to or less than 2.0dB/100 turns.

As can be seen from the result of comparison between FIGS. 1 and 3, therelative refractive index difference Δ₁ of the SMF according to thefirst embodiment having a delay time (4.876 μs/km) equal to that of acutoff shifted fiber according to the related art varies at a part inwhich the relative refractive index difference Δ₁ increases with anincrease of the radius a₂ (that is, a part in which the relativerefractive index difference Δ₁ moves away from 0 and a lower part in thegraph of FIG. 3). The radius a of the SMF according to the firstembodiment having a delay time equal to that of the cutoff shift fiberis expressed by Equation (4) using functions κ₀(a₁), κ₁(a₁) and κ₂(a₁)of the relative refractive index difference Δ₁ and the radius a₁. Inthis specification, a function refers to a fitting function andarguments thereof may be omitted.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack & \; \\{a_{2} = \frac{K_{0}\left( a_{i} \right)}{\left\{ {{K_{1}\left( a_{1} \right)}^{2} - \left( {\Delta_{1} - {K_{1}\left( a_{1} \right)}} \right)^{2}} \right\}^{K_{2}{(a_{1})}}}} & (4)\end{matrix}$

As described above, when the radius a₁ of the SMF according to the firstembodiment is 1.0 μm, the value of the function κ₀(a₁) is 2.00, thevalue of the function κ₁(a₁) is −1.42 and the value of the functionκ₂(a₁) is 0.50. As shown in FIG. 3, in a boundary (a solid line (D) inFIG. 3) of the SMF in which the MFD is equal to or greater than 11.5 μm,the radius a₂ when the relative refractive index difference Δ₁ is 0.0%is 5.56 μm.

With an increase of the radius a₂, the influence of the first clad onthe MFD decreases and the refractive index distribution of the SMFapproaches that of the step index type. In this case, the MFD isdetermined according to only a structure (that is, parameters such asthe radius at or the refractive index n₁) of a core. Accordingly, therelative refractive index difference Δ₁ converges on a value dependingon the radius a₁ with an increase of the radius a₂. Accordingly, theradius a₂ at which the MFD of the SMF according to the first embodimentis 11.5 μm is expressed by Equation (5) using a function of κ₃(a₁) ofthe relative refractive index difference Δ₁ and the radius a₁ and afunction κ₄(a₁) in which the relative refractive index difference Δ₁converges with an increase of the radius a₂.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack & \; \\{a_{2} = {5.56 + {{K_{3}\left( a_{1} \right)}{\log \left( {1 - \frac{\Delta_{1}}{K_{4}\left( a_{1} \right)}} \right)}}}} & (5)\end{matrix}$

When the radius a₁ of the SMF according to the first embodiment is 1.0μm, the value of the function κ₃(a₁) is −3.94 and the value of thefunction κ₄(a₁) is −0.88. As shown in FIG. 3, in a boundary (the solidline (E) in FIG. 3) of the SMF in which the bending loss is equal to orless than 2.0 dB/100 turns, when the relative refractive indexdifference Δ₁ is 0.0%, the radius a₂ is 7.68 μm.

When the refractive index distribution of the SMF approaches a simplestep index type with an increase of the radius a₂ as described above,the bending loss is also determined according to only the structure ofthe core. Since the relative refractive index difference Δ₁ converges onthe value determined according to the radius a₁ with an increase of theradius a₂, the radius a₂ in which the bending loss SMF is 2.0 dB/100turns is expressed by Equation (6) using functions of κ₅(a₁) and κ₆(a₁)of the relative refractive index difference Δ₁ and the radius a₁.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack & \; \\{a_{2} = {7.68 + {{K_{5}\left( a_{1} \right)}{\log \left( {1 - \frac{\Delta_{1}}{K_{6}\left( a_{1} \right)}} \right)}}}} & (6)\end{matrix}$

When the radius a₁ of the SMF according to the first embodiment is 1.0μm, the value of the function κ₅(a₁) is −1.08 and the value of theκ₆(a₁) is −0.82.

A hatched part in FIG. 3 represents a design area of the SMF accordingto the first embodiment which satisfies the prescriptions ofRecommendation G654.D and which can realize a delay time (4.876 μs/km)equal to or less than that of the cutoff shifted fiber.

As described above, a selection range of the radius a₂ with respect tothe relative refractive index difference Δ₁ is limited by the conditionsof the delay time, the MFD, and the bending loss. By employing thestructure corresponding to the hatched part of FIG. 3, the SMF accordingto the first embodiment realizes optical characteristics equal to thoseof a cutoff shift fiber according to the related art and realizes adelay time equal to or less than the delay time of the cutoff shiftfiber. As the relative refractive index difference Δ₁ decreases, thedelay time of the SMF according to the first embodiment decreases.

A relationship between the radius a₂ and the relative refractive indexdifference Δ₁ in the SMF according to the first embodiment of theinvention in which the radius a₁ is 1.5 μm is shown in FIG. 4. Solidlines (D) and (E) in FIGS. 4, 5 and FIGS. 9 to 11 which will bedescribed later represent the same details as the solid lines (D) and(E) in FIG. 3. A boundary of a selection range of the radius a₂ withrespect to the relative refractive index difference Δ₁ in the SMF inwhich the delay time is equal to or less than the delay time of a cutoffshifted fiber is expressed by Equation (4). A boundary of a selectionrange of the radius a₂ with respect to the relative refractive indexdifference Δ₁ in the SMF in which the MFD is equal to or greater than11.5 μm is expressed by Equation (5). A boundary of a selection range ofthe radius a₂ with respect to the relative refractive index differenceΔ₁ in the SMF in which the bending loss is equal to or less than 2.0dB/100 turns is expressed by Equation (6). When the radius a₁ is 1.5 μm,the value of the function κ₀(a₁) is 2.00, the value of function κ₁(a₁)is −0.86, the value of the function κ₂(a₁) is 0.43, the value of thefunction κ₃(a₁) is −3.94 and the value of the function κ₄(a₁) is −0.50.The hatched part in FIG. 4 represents a design area of the SMF accordingto the first embodiment satisfying the prescriptions of RecommendationG654.D.

A relationship between the radius a₂ and the relative refractive indexdifference Δ₁ of the first clad in the SMF according to the firstembodiment of the invention in which the radius a₁ of the core area is2.0 μm is shown in FIG. 5. A boundary of a structure in which the delaytime is equal to or less than the delay time of the cutoff shift fiberis expressed by Equation (4) described above. A boundary of a structurein which the MFD is equal to or greater than 11.5 μm is expressed byEquation (5) described above. A boundary of a structure in which thebending loss is equal to or less than 2.0 dB/100 turns is expressed byEquation (6). When the radius a₁ of the core area is 2.0 μm, the valueof the function κ₀(a₁) is 2.00, the value of function κ₁(a₁) is −0.50,the value of the function κ₂(a₁) is 0.38, the value of the functionκ₃(a₁) is −3.94 and the value of the function κ₄(a₁) is −0.36. Thehatched part in FIG. 5 represents a design area of the SMF satisfyingthe prescriptions of Recommendation G654.D of the ITU-T by thesingle-mode optical fiber according to the invention.

As shown in FIGS. 3 to 5, the relative relationship between the relativerefractive index difference Δ₁ and the radius a₂ varies depending on theradius a₁. The design area of the SMF according to the first embodimentis specified by expressing the coefficients of the functionsrepresenting the boundaries surrounding the hatched parts in FIGS. 3 to5 as the functions of the radius at. The invention is based on thepremise that the radius a₁ is equal to or greater than 1.0 μm inconsideration of conditions for easily manufacturing the optical fiber.

In the boundary (that is, the boundary expressed by Equation (1)) of thestructure in the SMF in which the delay time equal to or less than thedelay time of the cutoff shift fiber is achieved, the value of thefunction κ₀(a₁) is 2.00 regardless of the radius at of the core area. Arelationship between the values of the functions κ₁(a₁) and κ₂(a₁) andthe radius a₁ is shown in FIG. 6. As shown in FIG. 6, the functionκ₁(a₁) is expressed by Equation (7) and the function κ₂(a₁) is expressedby Equation (8).

[Equation 7]

K ₁=−1.43a ₁ ^(−1.45)  (7)

[Equation 8]

K ₂=0.50a ₁ ^(−0.37)  (8)

By expressing the boundary of the structure in which the delay time is4.876 μs/km using the radius a₁ the relative refractive index differenceΔ₁ and the radius a₂, the design area (that is, an area representingoptical characteristics) of the SMF according to the first embodiment isexpressed by Equation (1).

[Equation 9]

a ₂≥2{(1.43a ₁ ^(−1.45))²−(Δ₁+1.43a ₁ ^(−1.45))²}^(−0.50a) ¹^(−6.37)  (1)

In the boundary (the boundary expressed by Equation (5)) of thestructure in which the MFD is equal to or greater than 11.5 μm, thevalue of the function κ₃(a₁) is −3.94 regardless of the radius a₁. Arelationship between the value of the function κ₄(a₁) and the radius a₁is shown in FIG. 7. As shown in FIG. 7, the function κ₄(a₁) is expressedby Equation (10).

[Equation 10]

K ₄=−0.19−0.69a ₁ ^(−2.00)  (10)

By expressing the boundary of the structure in which the MFD is 11.5 μmusing the radius a₁, the relative refractive index difference Δ₁ and theradius a₂, the design area of the SMF according to the first embodimentis expressed by Equation (11).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 11} \right\rbrack & \; \\{a_{2} \geq {5.56 - {3.94\; {\log \left( {1 + \frac{\Delta_{1}}{0.19 + {0.69a_{1}^{- 2.00}}}} \right)}}}} & (11)\end{matrix}$

Regarding the boundary of the structure (the boundary expressed byEquation (6)) in which the bending loss is equal to or less than 2.0dB/100 turns, a relationship between the values of the functions κ₅(a₁)and κ₆(a₁) and the radius a₁ is shown in FIG. 8. As shown in FIG. 8, thefunction κ₅(a₁) is expressed by Equation (12) and the function κ₆(a₁) isexpressed by Equation (13).

[Equation 12]

K ₅=1.14−2.51a ₁  (12)

[Equation 13]

K ₆=−0.81a ₁ ^(−0.77)  (13)

By expressing the boundary of the structure in which the bending loss is2.0 dB/100 turns using the radius at, the relative refractive indexdifference Δ₁ and the radius a₂, the design area of the SMF according tothe first embodiment is expressed by Equation (9).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack & \; \\{a_{2} \leq {7.68 + {\left( {1.14 - {2.51a_{1}}} \right)\; {\log \left( {1 + \frac{\Delta_{1}}{0.81a_{1}^{- 0.77}}} \right)}}}} & (9)\end{matrix}$

Based on the above description, the SMF according to the firstembodiment includes a core of which the radius a₁ is equal to or greaterthan 1.0 μm and equal to or less than 4.3 μm, and satisfies Equation (1)described above and Equation (2) described below as the relationshipbetween the radius a₂ and the relative refractive index difference Δ₁.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack} & \; \\{{5.56 - {3.94\; {\log \left( {1 + \frac{\Delta_{1}}{0.19 + {0.69\; a_{1}^{- 2.00}}}} \right)}}} \leq a_{2} \leq {7.68 + {\left( {1.14 - {2.51\; a_{1}}} \right){\log \left( {1 + \frac{\Delta_{1}}{0.81\; a_{1}^{- 0.77}}} \right)}}}} & (2)\end{matrix}$

When the radius a₁ and the radius a₂ with respect to the relativerefractive index difference Δ₁ satisfy the above-mentioned conditions,the SMF according to the first embodiment realizes the same opticalcharacteristics as those of the cutoff shifted fiber and realizes thedelay time which is equal to or less than the delay time of the cutoffshifted fiber.

Table 1 shows an example of design parameters of the SMF according tothe first embodiment. In the design shown in Table 1, the same opticalcharacteristics as those of the cutoff shift fiber are achieved and areduction in the delay time of 0.05 μs/km is achieved. By using the SMFdesigned as described above, a reduction in delay time of about 1 ms isachieved in a long-distance network of which the network length amountsto about 10000 km such as a submarine optical cable crossing the PacificOcean. The design parameters shown in Table 1 are an example satisfyingthe above-mentioned conditions and the same advantageous effects as inthe SMF having the design parameters shown in Table 1 are achieved bythe SMF having the structure satisfying the above-mentioned conditions.

TABLE 1 Structure parameter Optical characteristics a₁ 1.0 μm MFD@λ =1.55 μm 11.8 μm a₂ 8.5 μm λ_(c) 1.53 μm Δ₁ −0.70% α_(b)@1.625 μm 1.2dB/100 turns R = 30 mm Δ₂ −0.17% GD@1.55 μm 4.847 μs/km

Second Embodiment

A relationship between the radius a- and the relative refractive indexdifference Δ₁ when the radius a₁ is 1.0 μm and a design area of an SMFaccording to a second embodiment of the invention satisfiesRecommendation G654.E is shown in FIG. 9. In consideration of therelative refractive index difference Δ₂ and the radius a₂, the cutoffwavelength of the SMF according to the second embodiment is set to beequal to or less than 1.53 μm. A solid line (D) in FIG. 9 represents aboundary of a structure in which the MFD is 11.5 μm similarly to thefirst embodiment. A solid line (E) in FIG. 9 represents a boundary of astructure in which the bending loss is equal to or less than 0.1 dB/100turns similarly to the first embodiment. A hatched part in FIG. 9represents a design area in which all the above-mentioned conditionswith respect to the cutoff wavelength, the MFD, and the bending loss aresatisfied and the delay time (equal to or less than 4.876 μs/km) whichis equal to or less than the delay time of the cutoff shifted fiber canbe realized.

When the radius a₁ is 1.0 μm, the boundary in which the delay time isequal to or less than the delay time of the cutoff shifted fiber isexpressed by Equation (4) similarly to the first embodiment. The valuesof the functions κ₀(a₁), κ₁(a₁) and κ₂(a₁) are equal to the values ofthe functions κ₀(a₁), κ₁(a₁), and κ₂(a₁) described above in the firstembodiment. The boundary in which the MFD is equal to or greater than11.5 μm is expressed by Equation (5). The values of the functions κ₃(a₁)and κ₄(a₁) are equal to the values of the functions κ₃(a₁) and κ₄(a₁)described above in the first embodiment.

In the boundary of the structure in which the bending loss is equal toor less than 0.1 dB/100 turns, when the relative refractive indexdifference Δ₁ is 0.0%, the radius a₂ is 6.88 μm. As the radius a₂increases, the influence of the first clad on the bending loss decreasesand the bending loss is determined according to only the structure ofthe core area. With the increase of the radius a₂, the relativerefractive index difference Δ₁ converges on a value which is determinedaccording to the radius a₁ of the core area. Accordingly, the radius a₂at which the bending loss is 0.1 dB/100 turns in the SMF according tothe invention is expressed by Equation (14) using the functions κ₇(a₁)and κ₈(a₁) Of the relative refractive index difference Δ₁ and the radiusa₁.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack & \; \\{a_{2} = {6.68 - {{K_{7}\left( a_{1} \right)}{\log \left( {1 - \frac{\Delta_{1}}{K_{8}\left( a_{1} \right)}} \right)}}}} & (14)\end{matrix}$

When the radius a₁ is 1.0 μm, the value of the function κ₇(a₁) is −3.74and the value of the function κ₈(a₁) is −1.36. The design area in whichthe relative refractive index difference Δ₁ is high is limited by anincrease of the delay time. Design ranges of the MFD and the radius a₂are limited by required conditions of the bending loss. As shown in FIG.9, the delay time decreases as the relative refractive index differenceΔ₁ decreases.

A relationship between the radius a and the relative refractive indexdifference Δ₁ when the radius a₁ is 1.5 μm and a design area of the SMFaccording to the second embodiment satisfying Recommendation G654.E isshown in FIG. 10. Even when the radius a₁ is 1.5 μm, the boundary inwhich the delay time is equal to or less than the delay time of thecutoff shift fiber is expressed by Equation (4). The values of thefunctions κ₀(a₁), κ₁(a₁) and κ₂(a₁) are equal to the values of thefunctions κ₀(a₁), κ₁(a₁), and κ₂(a₁) described above in the firstembodiment. The boundary in which the MFD is equal to or greater than11.5 μm is expressed by Equation (5). The values of the functions κ₁(a₁)and κ₄(a₁) are equal to the values of the functions κ₃(a₁) and κ₄(a₁)described above in the first embodiment.

The boundary of the structure in which the bending loss is equal to orless than 0.1 dB/100 turns is expressed by Equation (14). Since theradius a₁ is 1.5 μm, the value of the function κ₇(a₁) is −5.60 and thevalue of the function κ₈(a₁) is −0.87. The hatched part in FIG. 10represents a design area in which all the above-mentioned conditionswith respect to the cutoff wavelength, the MFD and the bending loss aresatisfied and the delay time equal to or less than the delay time of thecutoff shifted fiber is achieved.

A relationship between the radius a₂ and the relative refractive indexdifference Δ₁ when the radius a₁ is 2.0 μm and a design area of the SMFaccording to the second embodiment satisfying Recommendation G654.E isshown in FIG. 11. Even when the radius a₁ is 2.0 μm, the boundary inwhich the delay time is equal to or less than the delay time of thecutoff shifted fiber is expressed by Equation (4). The values of thefunctions κ₀(a₁), κ₁(a₁) and κ₂(a₁) are equal to the values of thefunctions κ₀(a₁), κ₁(a₁), and κ₂(a₁) described above in the firstembodiment. The boundary in which the MFD is equal to or greater than11.5 μm is expressed by Equation (5). The values of the functions κ₃(a₁)and κ₄(a₁) are equal to the values of the functions κ₃(a₁) and κ₄(a₁)described above in the first embodiment.

The boundary of the structure in which the bending loss is equal to orless than 0.1 dB/100 turns is expressed by Equation (14). Since theradius a₁ is 2.0 μm, the value of the function κ₇(a₁) is −6.25 and thevalue of the function κ₈(a₁) is −0.61. The hatched part in FIG. 11represents a design area in which all the above-mentioned conditionsassociated with the cutoff wavelength, the MFD and the bending loss aresatisfied and the delay time equal to or less than the delay time of thecutoff shift fiber is achieved.

As shown in FIGS. 9 to 11, the relative relationship between therelative refractive index difference Δ₁ and the radius a₂ depends on theradius a₁. The boundaries in which the MFD is equal to or greater than11.5 μm and which are shown in FIGS. 9 to 11 are the same as theboundaries which are shown in FIGS. 3 to 5. The structure of the SMF inwhich the MFD is equal to or greater than 11.5 μm is expressed byEquation (9). The boundaries in which the delay time is equal to or lessthan the delay time of the cutoff shift fiber and which are shown inFIGS. 9 to 11 are the same as the boundaries which are shown in FIGS. 3to 5. The structure of the SMF in which the delay time is equal to orless than the delay time of the cutoff shift fiber is expressed byEquations (9) and (11).

The boundary of the structure in which the bending loss is equal to orless than 0.1 dB/100 turns is expressed by Equation (14). Variations ofthe function κ₇ (the left axis in FIG. 12) and the function κ₈ (theright axis in FIG. 12) with respect to the radius a₁ are shown in FIG.12. As shown in FIG. 12, the function κ₇(a₁) is expressed by Equation(15) and the function κ₈(a₁) is expressed by Equation (16).

[Equation 17]

K ₇ =a ₁(−4.28+0.51a ₁)  (15)

[Equation 18]

K ₈=−1.36a ₁ ^(−1.12)  (16)

As described above, the design area of the radius a₂ of the SMFaccording to the second embodiment in which the bending loss is equal toor less than 0.1 dB/100 turns is expressed by Equation (17).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack & \; \\{a_{2} \leq {6.68 + {{a_{1}\left( {{- 4.28} + {0.51\; a_{1}}} \right)}{\log \left( {1 + \frac{\Delta_{1}}{1.36\; a_{1}^{- 1.12}}} \right)}}}} & (17)\end{matrix}$

As described above, the SMF according to the second embodiment includesa core in which the radius a₁ is equal to or greater than 1.0 μm andequal to or less than 4.3 μm and includes a design area in whichEquations (1) and (2) are satisfied as the relationship between theradius a₂ and the relative refractive index difference Δ₁.

$\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack} & \; \\{\mspace{85mu} {a_{2} \geq {2\left\{ {\left( {1.43a_{1}^{- 1.45}} \right)^{2} - \left( {\Delta_{1} + {1.43a_{1}^{- 1.45}}} \right)^{2}} \right\}^{\text{?}}}}} & (1) \\{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack} & \; \\{{5.56 - {3.94\; {\log \left( {1 + \frac{\Delta_{1}}{0.19 + {0.69a_{1}^{- 2.00}}}} \right)}}} \leq a_{2} \leq {6.68 + {{a_{1}\left( {{- 4.28} + {0.51\; a_{1}}} \right)}{\log \left( {1 + \frac{\Delta_{1}}{1.36a_{1}^{- 1.12}}} \right)}}}} & (2) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

When the radius a₁ and the radius a₂ with respect to the relativerefractive index difference Δ₁ satisfy the above-mentioned conditions,the SMF according to the second embodiment achieves the same opticalcharacteristics as the cutoff shift fiber and achieves a delay timeequal to or less than the delay time of the cutoff shift fiber.

In FIG. 13, when the radius at of the core area is 1.0 μm and when theradius at of the core area is 2.0 μm, boundaries in which the cutoffwavelength is equal to or less than 1.53 μm with respect to the radiusa₂ and the relative refractive index difference Δ₂ when the relativerefractive index difference Δ₁ is set to −0.1% and the relativerefractive index difference Δ₁ is set to −0.80% are shown. In a part inwhich the relative refractive index difference Δ₂ is lower than theboundaries (that is a part in which the relative refractive indexdifference Δ₂ approaches 0 and an upper part in the graph which is shownin FIG. 13), the above-mentioned required conditions with respect to thecutoff wavelength, the MFD and the bending loss are satisfied. As shownin FIG. 9, the boundary in which the cutoff wavelength is equal to orless than 1.53 μm varies depending on the radius a₁ of the core and therelative refractive index difference Δ₁. Accordingly, the relativerefractive index difference Δ₂ in which the cutoff wavelength is 1.53 μmis expressed by the radius a₁, the relative refractive index differenceΔ₁ and the radius a₂.

When the radius a₂ increases, the boundary in which the cutoffwavelength is 1.53 μm varies to a part in which the relative refractiveindex difference Δ₂ is low. When the radius a₂ increases, an influenceof the structure of the core on the cutoff wavelength decreases and therelative refractive index difference Δ₂ converges regardless of theradius a₁ and the relative refractive index difference Δ₁. Theconverging value of the relative refractive index difference Δ₂ is−0.033%. When the converging value of the relative refractive indexdifference Δ₂ and the functions κ₉(a₁,Δ₁) and κ₁₀(a₁,Δ₁) with the radiusa₁ and the relative refractive index difference Δ₁ as variables, therelative refractive index difference Δ₂ in which the cutoff wavelengthis 1.53 μm is expressed by Equation (20).

[Equation 20]

Δ₂=−0.033+K ₉(a ₁,Δ₁)a ₂ ^(K) ¹⁰ ^((a) ¹ ^(,Δ) ¹ ⁾  (20)

Variations of the function κ₉ (the left axis in FIG. 14) and thefunction κ₁₀ (the right axis in FIG. 14) with respect to the relativerefractive index difference Δ₁ when the radius a₁ is set to 1.0 μm, 2.0μm and 4.3 μm are shown in FIG. 14. When the relative refractive indexdifference Δ₁ is 0.0% as described above, the relationships between therelative refractive index difference Δ₁ and the functions κ₉ and κ₁₀ areexpressed by Equations (21) and (22) using the functions κ₁₁(a₁),κ₁₂(a₁) and κ₁₃(a₁) having the radius at as a variable.

[Equation 23]

K ₉=−7.490+K ₁₁(a ₁)Δ₁ +K ₁₂(a ₁)Δ₁ ²  (21)

[Equation 24]

K ₁₀=−1.876+K ₁₃(a ₁)Δ₁  (22)

When the radius at is 4.3 μm and the relative refractive indexdifference Δ₁ is equal to or less than −0.531%, the cutoff wavelength isequal to or greater than 1.53 μm regardless of the radius a₂ and therelative refractive index difference Δ₂. Accordingly, when the radius atis 4.3 μm, Equations (21) and (22) are applied to a structure of the SMFin which the relative refractive index difference Δ₁ is equal to orhigher than −0.531%.

Variations of the function κ₁₁ (the left axis in FIG. 15) and thefunction κ₁₂ (the right axis in FIG. 15) with respect to the radius atare shown in FIG. 15. When the radius a₁ is 0.0 μm, both the functionsκ₁₁ and κ₁₂ lose dependency on the relative refractive index differenceΔ₁ and thus the functions κ₁₁ and κ₁₂ are 0 (zero). The curve which isshown as a solid line in FIG. 15 is expressed by Equation (23). Thecurve which is shown as a dotted line in FIG. 15 is expressed byEquation (24).

[Equation 25]

K ₁₁=−0.187a ₁ ^(3.407)  (23)

[Equation 26]

K ₁₂=−0.044a ₁ ^(4.324)  (24)

A variation of the function κ₁₃ with respect to the radius a₁ is shownin FIG. 16. When the radius a₁ is 0.0 μm, the function κ₁₃ losesdependency on the relative refractive index difference Δ₁ and thus thefunction κ₁₃ is 0 (zero). The curve which is shown as a solid line inFIG. 16 is expressed by Equation (25).

[Equation 27]

K ₁₃=−0.014a ₁ ^(3.099)  (25)

Based on the above description, by expressing the boundary of thestructure of the SMF according to the second embodiment in which thecutoff wavelength is equal to or less than 1.53 μm using the radius a₁,the relative refractive index difference Δ₁ and the radius a₂, therelative refractive index difference Δ₂ is expressed by Equation (3).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack & \; \\{{\Delta_{2} \geq {{- 0.033} + {\left( {{- 7.490} - {0.187a_{1}^{3.407}\Delta_{1}} - {0.044a_{1}^{4.324}\Delta_{1}^{2}}} \right)a_{2}^{{- 1.876} - {0.014\text{?}}}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (3)\end{matrix}$

The above-mentioned SMF according to the second embodiment is designedsuch that the radii a₁ and a₂ and the relative refractive indexdifferences Δ₁ and Δ₂ satisfy the above-mentioned suitable conditions.

In the invention, by performing design such that the radius a₁ is in arange of equal to or greater than 1.0 μm and equal to or less than 4.3μm, suitable design areas of the radius a₂ and the relative refractiveindex difference Δ₁ which are expressed by Equations (1) and (2) arespecified in the SMF based on Recommendation G654.D. Similarly, in theSMF based on Recommendation G654.E, suitable design areas of the radiusa₂ and the relative refractive index difference Δ₁ which are expressedby Equations (18) and (19) are specified. In the suitable areas, theachievable delay time of the SMF is roughly determined according to therelative refractive index difference Δ₁ and increases as the relativerefractive index difference Δ₁ decreases. By appropriately selecting theradii a₁ and a₂ and the relative refractive index difference Δ₁ in theabove-mentioned suitable areas, the cutoff wavelength satisfies theconditions which are required to apply the SMF to a long-distancecommunication network and the relative refractive index difference Δ₂ isdetermined according to Equation (3).

The second clad of the SMF according to the second embodiment is notlimited to the second clad with the refractive index distribution whichis shown in FIG. 2. An example of a refractive index distribution inwhich a low-index area in which the refractive index is defined as n₄and a relative refractive index difference from the second clad isdefined as Δ₃ is provided in an area with a width w from a radius a₃ toa radius a₄ of the second clad which is shown in FIG. 2 is shown in FIG.17. That is, the radius a₃ is greater than the radius a₂ and therefractive index n₄ is lower than the refractive index n₃. It is knownthat the low-index area of the second clad relaxes trade-off which iscaused in the optical characteristics such as the MFD or the bendingloss. With the SMF according to the second embodiment, the design areais expected to be suitably enlarged. By forming a hole instead of thelow-index area in the second clad, the same advantageous effects as inthe SMF including the low-index area in the second clad are obtained.

Third Embodiment

A configuration of an optical transmission system 100 according to theinvention is shown in FIG. 18. The optical transmission system 100includes a transmitter 102, the optical fiber (SMF) 104 according to theinvention and a receiver 106. The transmitter 102 and the optical fiber104 are connected to each other via a connector. The optical fiber 104and the receiver 106 are connected to each other via a connector. Sincethe optical transmission system 100 includes the optical fiber 104, atransmission delay of the optical transmission system 100 decreases.Accordingly, the optical transmission system 100 can respond to a demandfor a reduction in delay time between the transmitter 102 and thereceiver 106.

Fourth Embodiment

Structures of optical fibers 51, 52, 53 and 54 which aresingle-conductor optical fibers and in which a plurality of cores aredisposed in a sectional view crossing the longitudinal direction areshown in FIGS. 19A, 19B, 19C and 19D. One or more cores of the opticalfibers 51, 52, 53 and 54 are low-delay cores which satisfy theconditions described above in the first embodiment or the secondembodiment and which can reduce a delay time of the optical fiber.

A sectional surface of an optical fiber in which only four low-delaycores 60A are disposed in a second clad 66 (that is, the second clad inthe first embodiment and the second embodiment) with a diameter of 125μm is shown in FIG. 19A. Each low-delay core 60A includes a core (afirst core) 62 at the center in a sectional view and a first clad 64that is disposed on an outer circumferential portion of the core 62. Asectional surface of an optical fiber in which a core (a second core)60B is disposed at the center of a second clad 66 with an outer diameterof 125 μm and four low-delay cores 60A are disposed on a concentriccircle centered on the core 60B is shown in FIG. 19B. A sectionalsurface of an optical fiber including a second clad 66 with a diameterof 175 μm and a coating 70 with a diameter of 250 μm that is disposed onan outer circumferential portion of the second clad 66 is shown in FIG.19C. In the SMF which is shown in FIG. 19C, a low-delay core 60A isdisposed at the center of the second clad 66 and cores (third cores) 60Care disposed on a concentric circle centered on the low-delay core 60A.A sectional surface of an optical fiber in which a low-delay core 60A isdisposed at the center of a second clad 66 with a diameter of 250 μmwhich is thought to be a maximum diameter of the clad with whichreliability can be secured and a plurality of cores (fourth cores) 60Dare disposed in a hexagonal shape with intervals interposed therebetween(that is, packed most closely) in the second clad outside the low-delaycore 60A is shown in FIG. 19D.

With the optical transmission system including the optical fibers 51,52, 53 and 54 which are shown in FIGS. 19A to 19D, the opticalcommunication according to the related art and optical communicationwith a reduced delay time can be simultaneously and suitably realized.

Fifth Embodiment

An optical transmission system 200 including the optical fiber accordingto the invention is shown in FIG. 20. The optical transmission system200 includes an SMF (an optical fiber) 150, a plurality of transmitters172 and a plurality of receivers 174. The SMF 150 may be one of the SMFsaccording to the first to third embodiments and the optical fibers 51,52, 53 and 54 according to the fourth embodiment. At least one of theplurality of transmitters 172 is coupled to one end of the SMF 150 via afan-in device. At least one of the plurality of receivers 174 is coupledto the other end of the SMF 150 via a fan-in device. With the opticaltransmission system 200, the optical communication according to therelated art and the optical communication with a decreased transmissiondelay can be simultaneously realized. Accordingly, the opticaltransmission system 200 is capable of flexibly responding to a demandfor a reduction of delay time in a transmission line.

(Preferable Design Conditions of the Optical Fiber)

A result of numerical calculation of the relationship between the radiusat and the relative refractive index difference Δ₁ with respect to aRayleigh scattering loss α_(R) due to an influence of the first clad onan electric field distribution of light which is transmitted by the coreof the SMF when the MFD is 11.5 μm is shown in FIG. 21. A result ofnumerical calculation of the relationship between the radius at and therelative refractive index difference Δ₁ with respect to the Rayleighscattering loss α_(R) due to an influence of the first clad on theelectric field distribution of light which is transmitted by the core ofthe SMF when the MFD is 15.0 μm is shown in FIG. 22. As shown in FIGS.21 and 22, it can be seen that the Rayleigh scattering loss α_(R) of theSMF increases by decreasing the radius a₁ and decreasing the relativerefractive index difference Δ₁.

A relationship between the Rayleigh scattering loss α_(R), the MFD, thegroup delay time of light which is transmitted by the core per unitlength, and the radius a₁ in the SMF based on Recommendation G654.Dbased on the results of numerical calculation which are shown in FIGS.21 and 22 is shown in FIG. 23. A relationship between the Rayleighscattering loss α_(R), the MFD, the group delay time of light which istransmitted by the core per unit length, and the radius a₁ in the SMFbased on Recommendation G654.E based on the results of numericalcalculation which are shown in FIGS. 21 and 22 is shown in FIG. 24.

The Rayleigh scattering loss α_(R) of a conventional SMF is consideredto be about 0.17 dB/km. In consideration that the group delay time ofthe cutoff shift fiber (CSF) is about 4.877 μs/km, the opticalcharacteristics based on Recommendation G654.D and Recommendation G654.Eand low loss and low delay time can be simultaneously realized and amore suitable design area of the optical fiber according to theinvention can be provided, by setting the radius at and the relativerefractive index difference Δ₁ corresponding to the hatched parts inFIGS. 23 and 24. The hatched parts in FIGS. 23 and 24 represent a rangein which the boundary in which the MFD is equal to or greater than 11.5μm and the boundary in which the Rayleigh scattering loss Δ_(R) is 0.17dB/km overlap each other toward a part in which the relative refractiveindex difference Δ₁ is lower than the boundaries (that is, a part inwhich the relative refractive index difference Δ₁ approaches 0 and anupper part in the graph which is shown in FIG. 13) such that therecommendations are satisfied. That is, in the above-mentionedembodiments, by additionally including conditions in which the Rayleighscattering loss α_(R) is equal to or less than 0.17 dB/km, the opticalcharacteristics based on Recommendation G654.D and Recommendation G654.Eand the low loss and low delay time can be simultaneously realized and amore suitable design area can be provided.

EXAMPLES

An SMF (an optical fiber) according to the invention was manufactured bytrial in consideration of the design area, the relative relationshipbetween parameters and the suitable conditions which have been describedabove in the embodiments. A refractive index distribution of thetrial-manufactured SMF is shown in FIG. 25. Evaluation ofcharacteristics of a conventional SMF and a CSF along with thetrial-manufactured SMF according to the invention was carried out.

A wavelength serving in measuring an MFD, an effective sectional areaA_(eff), a bending loss per winding with a bending radius of 15 mm, atransmission loss, a Rayleigh scattering loss, a chromatic dispersionand a nonlinear coefficient, measurement results of each opticalproperty and measurement methods used for various measurements in theoptical fiber according to the invention which was manufactured by trial(hereinafter also referred to as a trial-manufactured optical fiber) areshown in FIG. 26. For the purpose of comparison, the same measurement asin the trial-manufactured optical fiber according to the invention (a“trial-manufactured optical fiber” in FIG. 26) was performed on theconventional SMF and the cutoff shift fiber (CSF) based onRecommendation G654.D.

A structure of an optical fiber which was manufactured by trial in adesign area of structure parameters based on Recommendation G654.D inwhich characteristics of a cutoff shift fiber specified based on resultsof numerical calculation are prescribed is shown in FIG. 25. Thetrial-manufactured optical fiber includes a core which is formed of puresilica glass. The radius of the core is 1.0 μm and the radius of thefirst clad is 6.4 μm. The relative refractive index difference betweenthe core and the first clad is −0.38%, and the relative refractive indexdifference between the first clad and the second clad is −0.24%.

As shown in FIG. 26, as designed, the trial-manufactured optical fiberhas the equivalent MFD, the equivalent A_(eff), the equivalent cutoffwavelength, and the equivalent bending loss as in the CSF. Thetransmission loss of the trial-manufactured optical fiber is as low asin the conventional SMF. The wavelength dispersion and the nonlinearcoefficient of the trial-manufactured optical fiber have the equivalentvalues as in the CSF.

Results of measurement of wavelength characteristics of the MFD ofvarious fibers are shown in FIG. 27. In the trial-manufactured opticalfiber, the MFD is enlarged to the similar extent to that of the CSF inthe whole band from C-band to L-band. The tendency of enlargement of theMFD matches the results of numerical calculation represented by solidlines.

Results of measurement of a loss wavelength spectrum of various fibersare shown in FIG. 28. The shape of the loss wavelength spectrum of thetrial-manufactured optical fiber is the similar to the shapes of theconventional SMF and the CSF. The transmission loss of thetrial-manufactured optical fiber is equivalent to that of theconventional SMF in the whole range of measurement wavelengths.

Results of a plot with [wavelength λ]⁻⁴ in the loss wavelength spectrumwhich is shown in FIG. 28 are shown in FIG. 29. When the Rayleighscattering loss at the wavelength of 1.55 μm is analyzed based on theslopes of fitting lines in the range of from 0.52 μm⁻⁴ to 0.80 μm⁻⁴ ofλ⁻⁴, the Rayleigh scattering loss is 0.161 dB/km in thetrial-manufactured optical fiber, is 0.166 dB/km in the conventionalSMF, and is 0.146 dB/km in the CSF From these results, it is confirmedthat the Rayleigh scattering loss of the trial-manufactured opticalfiber is equivalent to that of the conventional SMF.

Measurement results of incident light power dependency of a phase shiftbased on a CW-SPM method, which correspond a nonlinear coefficientevaluation of various fibers are shown in FIG. 30. In the CW-SPM, thenonlinear coefficient (n₂/A_(eff)) can be analyzed from the slope of theincident light power dependency of a phase shift amount using Equation(26).

$\begin{matrix}{\Phi_{SPM} = {\frac{n_{2}}{A_{eff}}\frac{2\pi \; L_{eff}}{\lambda}P_{in}}} & (26)\end{matrix}$

In Equation (26), Φ_(SPM) represents a phase shift amount, λ representsa wavelength, L_(eff) represents an effective length of various fibers,and P_(in) represents power of incident light on various fibers. Thenonlinear coefficient which was analyzed based on the slope of thefitting line represented by a dotted line is 1.79×10⁻¹⁰/W in thetrial-manufactured optical fiber, 2.95×10⁻¹⁰/W in the conventional SMF,and 1.90×10⁻¹⁰/W in the CSF From these results, it is confirmed that thetrial-manufactured optical fiber has substantially the equivalent lownonlinearity to that in the CSF.

Measurement results of the group delay time based on an impulse responseapproach in various fibers are shown in FIG. 31. The horizontal axis ofthe graph which is shown in FIG. 31 represents the measured group delaytime in terms of a group delay time per unit length. In measurement, apulse width of a pulse which is emitted from a pulse light source wasmodulated in 100 ps and the length of the optical fiber to be measuredwas set to 350 m. In the CSF, the group delay time was reduced by 0.018μs/km in comparison with the conventional SMF. On the other hand, in thetrial-manufactured optical fiber, it is confirmed that the group delaytime was reduced additionally by 0.016 μs/km in comparison with that ofthe CSF.

Measurement results of wavelength dependency in the range from C-band toL-band of the measurement result of the group delay time which is shownin FIG. 31 are shown in FIG. 32. It is confirmed that the group delaytime of the trial-manufactured optical fiber was reduced by about 0.016μs/km from that of the CSF in the whole range from C-band to L-band.

The conventional SMF has a three-layered structure including a core, afirst clad, and a second clad, and the following parameters were usedfor the above-mentioned numerical calculation for the conventional SMF.

-   -   Radius of core: 3.5 μm    -   Radius of first clad: 6.5 μm    -   Relative refractive index difference between core and second        clad: 0.38%    -   Radius of second clad: 62.5 μm    -   Relative refractive index difference between first clad and        second clad: 0.05%    -   Refractive index of second clad (wavelength of 1.55 μm):        1.444377

The CSF has a three-layered structure including a core, a first clad,and a second clad and the following parameters were used for theabove-mentioned numerical calculation for the CSF

-   -   Radius of core: 6 μm    -   Refractive index of core (wavelength of 1.55 μm): 1.444377    -   Radius of first clad: 25 μm    -   Relative refractive index difference between first clad and        core: −0.35%    -   Radius of second clad: 62.5 μm    -   Relative refractive index difference between second clad and the        core: −0.25%

INDUSTRIAL APPLICABILITY

The invention is capable of being widely applied to optical fibers inareas of application in which a reduction of transmission delay ismainly required such as optical fibers for a long-distance communicationnetwork. The invention is capable of being applied to communicationbetween terminals in an optical communication system.

REFERENCE SIGNS LIST

-   -   60A: Low-delay core    -   60B: Core (second core)    -   60C: Core (third core)    -   60D: Core (fourth core)    -   62: Core (first core)    -   150: SMF (optical fiber)    -   172: Transmitter    -   174: Receiver    -   a₁, a₂, a₃: Radius    -   Δ₁, Δ₂, Δ₃: Relative refractive index difference

1. An optical fiber, comprising: a core; a first clad that is providedon an outer circumferential portion of the core and has a refractiveindex lower than that of the core; and a second clad that is provided onan outer circumferential portion of the first clad and has a refractiveindex lower than that of the first clad, wherein a mode field diameterat a wavelength of 1.55 μm is equal to or greater than 11.5 μm, whereina cutoff wavelength is equal to or less than 1.53 μm, wherein a bendingloss at a bending radius of 30 mm and a wavelength of 1.625 μm is equalto or less than 2.0 dB/100 turns, and wherein a delay time oftransmission light per unit length at a wavelength of 1.55 μm is equalto or less than 4.876 μs/km.
 2. The optical fiber according to claim 1,wherein a radius of the core is equal to or less than 1.0 μm and equalto or greater than 4.3 μm, and wherein a radius of the first cladsatisfies Equations (1) and (2): $\begin{matrix}{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack} & \; \\{\mspace{79mu} {a_{2} \geq {2\left\{ {\left( {1.43a_{1}^{- 1.45}} \right)^{2} - \left( {\Delta_{1} + {1.43a_{1}^{- 1.45}}} \right)^{2}} \right\}^{\text{?}}}}\;} & (1) \\{\mspace{79mu} {and}} & \; \\{\mspace{79mu} \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack} & \; \\{{5.56 - {3.94\; {\log \left( {1 + \frac{\Delta_{1}}{0.19 + {0.69a_{1}^{- 2.00}}}} \right)}}} \leq a_{2} \leq {7.68 + {\left( {1.14 - {2.51\; a_{1}}} \right){\log \left( {1 + \frac{\Delta_{1}}{0.81a_{1}^{- 0.77}}} \right)}}}} & (2) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$ where a₁ in Equations (1) and (2) represents the radius[μm] of the core, a₂ represents the radius [μm] of the first clad, andΔ₁ represents a relative refractive index difference [%] of the firstclad from the core.
 3. The optical fiber according to claim 2, whereinthe relative refractive index difference of the second clad from thefirst clad satisfies Equation (3): $\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack & \; \\{{\Delta_{2} \geq {{- 0.033} + {\left( {{- 7.490} - {0.187a_{1}^{3.407}\Delta_{1}} - {0.044a_{1}^{4.324}\Delta_{1}^{2}}} \right)a_{2}^{{- 1.876} - {0.014\text{?}}}}}},{\text{?}\text{indicates text missing or illegible when filed}}} & (3)\end{matrix}$ where Δ₂ represents the relative refractive indexdifference [%] of the second clad from the first clad.
 4. The opticalfiber according to claim 1 wherein, in a sectional view crossing alongitudinal direction, a plurality of low-delay cores including thecore as a first core and the first clad provided on an outercircumferential portion of the first core are disposed on concentriccircles at the center of the second clad.
 5. The optical fiber accordingto claim 4, wherein the core is disposed as a second core at the centerof the second clad.
 6. The optical fiber according to claim 1, wherein,in a sectional view crossing a longitudinal direction, a low-delay coreincluding the core as a first core and the first clad provided on anouter circumferential portion of the first core is disposed at thecenter of the second clad, and wherein the core is disposed as a thirdcore on a concentric circle at the center of the low-delay core.
 7. Theoptical fiber according to claim 1 wherein, in a sectional view crossinga longitudinal direction, a low-delay core including the core as a firstcore and the first clad provided on an outer circumferential portion ofthe first core is disposed at the center of the second clad, and whereinthe cores are packed most closely as fourth cores around the low-delaycore.
 8. An optical transmission system, comprising: the optical fiberaccording to claim 1; a transmitter that is connected to one end of theoptical fiber; and a receiver that is connected to the other end of theoptical fiber.